I have separated the Joint hierarchy in the file below, not as a suggestion of best practice, but to demo the function.

CV4_2023_drs_23_ANsm_01.c4d

Joints are nothing else than objects that move points around.
Polygon objects are based on Points; when those points are moved, we move the edges and, consequently, the polygons. As a result, the mesh gets deformed.

If two or even more Joints influence a certain area, it is typical that each influences the system a little bit differently. This is the weighting.

This must always be, with no exception, 100% in total. Sadly that is often not explained why, other than it needs to be 100%. If it is less than 100%, even the slightest bit, a moving forward caharter would leave these parts more or less behind. If it is more than 100%, those parts are moved fast forward.

We have sadly no movie younger than ten+ years that shows that.
https://www.cineversity.com/vidplaytut/weight_painting_part_1
I encourage presenters to show it. But well, Bret, the master (I mean it as I say it here!), does this with no effort after 05:30 [min: sec]

All the best